Problem 82 - Algebra and Trigonometry

Question

a. Graph \(y=e^{x}\) and \(y=1+x+\frac{x^{2}}{2}\) in the same viewing rectangle. b. Graph \(y=e^{x}\) and \(y=1+x+\frac{x^{2}}{2}+\frac{x^{3}}{6}\) in the same viewing rectangle. c. Graph \(y=e^{x}\) and \(y=1+x+\frac{x^{2}}{2}+\frac{x^{3}}{6}+\frac{x^{4}}{24}\) in the same viewing rectangle. d. Describe what you observe in parts (a)-(c). Try generalizing this observation.

Step-by-Step Solution

Answer

The Taylor series approximations become better as more terms are added - the graph of the expansion gets closer to the graph of the function itself.

Step 1: Graph the Functions in Part a

Start by graphing \(y=e^{x}\) and \(y=1+x+\frac{x^{2}}{2}\) together. Use different colours to distinguish between them.

Step 2: Graph the Functions in Part b

Similarly, graph \(y=e^{x}\) and \(y=1+x+\frac{x^{2}}{2}+\frac{x^{3}}{6}\) together.

Step 3: Graph the Functions in Part c

Finally, graph \(y=e^{x}\) and \(y=1+x+\frac{x^{2}}{2}+\frac{x^{3}}{6}+\frac{x^{4}}{24}\) together.

Step 4: Observe the Graphs

In this step, carefully observe how the graphs of the Taylor expansion become closer to the graph of \(y=e^{x}\) as more terms are added. Here, you're focusing on understanding how well the function can be approximated by Taylor series.

Step 5: Generalise the Observation

The final step involves making a general observation on the behavior of the graphs. From the previous steps, the graphs become more similar (or converge) as more terms in the Taylor series expansion are added. Therefore, the general observation is that a Taylor series with more terms provides a better approximation of the function.